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Any definition of a fraction should agree with the common ways of looking at a fraction. The three most common ways are:

1. Break an interval that is three units long into four equal pieces and take one of them:

2. Break a unit interval into four equal pieces and take three of them.

3. Divide four into three.

Related to the fraction $\dfrac 34$ is the ratio $3:4$. Two segments are said to have a ratio of $3:4$ if there is a common measure that can be "laid out" exactly 3 times onto the first segment and exactly 4 times onto the second segment.

The idea of a common measure is more primitive than the idea of a fraction. It was once thought that any two line segments had a common measure (were commensurable). We now know that isn't true. A side of a square and a diagonal of that square are not commensurable for example.

It would take more time than I'm willing to spend to show formally how to create the set, $\mathbb Q$, of rational numbers. The important properties are

Q1. $\dfrac ab = \dfrac cd$ if and only if $ad= bc$.

Q2. $\dfrac ab + \dfrac cd = \dfrac{ad+bc}{cd}$.

Q3. $\dfrac ab \cdot \dfrac cd = \dfrac{ac}{bd}$.

How do we justify property Q1? Assume for the moment that rational numbers behave the way we would like them to. If $n$ if a non zero integer, then we want $\dfrac nn$ to act like the number $1$. If that is the case, then we must have $\dfrac ab = \dfrac ab \dfrac nn = \dfrac{an}{bn}$. If we let $c=an$ and $d=bn$ then we find that $ad = bc$. Also we want $\dfrac 0n$ to act like $0$. In which case

\begin{align} \dfrac ab = \dfrac cd &\implies \dfrac{ad}{cd} - \dfrac{bc}{bd} = \dfrac{0}{cd} \\ &\implies \dfrac{ad-bc}{cd} = \dfrac{0}{cd} \\ &\implies ad-bc = 0 \\ &\implies ad = bc \end{align}

Finally, we want to embed the rational numbers $\mathbb Q$ into the set of real numbers $\mathbb R$. The simplest way is to just accept what we want to be true.

R1. For $x,y \in \mathbb R$ with $y \ne 0$ we define $\dfrac xy = x \cdot y^{-1}$.

Note that a lot of books define $y^{-1} = \dfrac 1y$. It isn't too hard to check that Q1, Q2, and Q3 are still true. So R1 is a logically consistent way to embed the rational number into the real numbers.

Any definition of a fraction should agree with the common ways of looking at a fraction. The three most common ways are:

1. Break an interval that is three units long into four equal pieces and take one of them:

2. Break a unit interval into four equal pieces and take three of them.

3. Divide four into three.

Related to the fraction $\dfrac 34$ is the ratio $3:4$. Two segments are said to have a ratio of $3:4$ if there is a common measure that can be "laid out" exactly 3 times onto the first segment and exactly 4 times onto the second segment.

The idea of a common measure is more primitive than the idea of a fraction. It was once thought that any two line segments had a common measure (were commensurable). We now know that isn't true. A side of a square and a diagonal of that square are not commensurable for example.

It would take more time than I'm willing to spend to show formally how to create the set, $\mathbb Q$, of rational numbers. The important properties are

Q1. $\dfrac ab = \dfrac cd$ if and only if $ad= bc$.

Q2. $\dfrac ab + \dfrac cd = \dfrac{ad+bc}{cd}$.

Q3. $\dfrac ab \cdot \dfrac cd = \dfrac{ac}{bd}$.

How do we justify property Q1? Assume for the moment that rational numbers behave the way we would like them to. If $n$ if a non zero integer, then we want $\dfrac nn$ to act like the number $1$. If that is the case, then we must have $\dfrac ab = \dfrac ab \dfrac nn = \dfrac{an}{bn}$. If we let $c=an$ and $d=bn$ then we find that $ad = bc$. Also we want $\dfrac 0n$ to act like $0$. In which case

\begin{align} \dfrac ab = \dfrac cd &\implies \dfrac{ad}{cd} - \dfrac{bc}{bd} = \dfrac{0}{cd} \\ &\implies \dfrac{ad-bc}{cd} = \dfrac{0}{cd} \\ &\implies ad-bc = 0 \\ &\implies ad = bc \end{align}

Finally, we want to embed the rational numbers $\mathbb Q$ into the set of real numbers $\mathbb R$. The simplest way is to just accept what we want to be true.

R1. For $x,y \in \mathbb R$ with $y \ne 0$ we define $\dfrac xy = x \cdot y^{-1}$.

Note that a lot of books define $y^{-1} = \dfrac 1y$. It isn't too hard to check that Q1, Q2, and Q3 are still true. So R1 is a logically consistent way to embed the rational number into the real numbers.

(9/24/2020) In response to @user599310's question.

You can "see" this for yourself with a sheet of lined paper and another sheet of paper (thin enough to see the lines on the other sheet of paper through it) with a segment drawn on it.

To break that segment into, say, $5$ equal pieces, select any $6$ consecutive lines on the lined sheet of paper and think of them as being numbered $0, 1,2,3,4$ and $5$.

Now place the sheet of paper with the segment on it so that one end of the segment touches line number $0$ and the other end of the segment touches line number $5$. (Such a thing can be constructed using a straight edge and a compass.)

Now the lines $1$ through $4$ break the segment into $5$ equal pieces.

Any definition of a fraction should agree with the common ways of looking at a fraction. The three most common ways are:

1. Break an interval that is three units long into four equal pieces and take one of them:

2. Break a unit interval into four equal pieces and take three of them.

3. Divide four into three.

Related to the fraction $\dfrac 34$ is the ratio $3:4$. Two segments are said to have a ratio of $3:4$ if there is a common measure that can be "laid out" exactly 3 times onto the first segment and exactly 4 times onto the second segment.

The idea of a common measure is more primitive than the idea of a fraction. It was once thought that any two line segments had a common measure (were commensurable). We now know that isn't true. A side of a square and a diagonal of that square are not commensurable for example.

Any definition of a fraction should agree with the common ways of looking at a fraction. The three most common ways are:

1. Break an interval that is three units long into four equal pieces and take one of them:

2. Break a unit interval into four equal pieces and take three of them.

3. Divide four into three.

Related to the fraction $\dfrac 34$ is the ratio $3:4$. Two segments are said to have a ratio of $3:4$ if there is a common measure that can be "laid out" exactly 3 times onto the first segment and exactly 4 times onto the second segment.

The idea of a common measure is more primitive than the idea of a fraction. It was once thought that any two line segments had a common measure (were commensurable). We now know that isn't true. A side of a square and a diagonal of that square are not commensurable for example.

It would take more time than I'm willing to spend to show formally how to create the set, $\mathbb Q$, of rational numbers. The important properties are

Q1. $\dfrac ab = \dfrac cd$ if and only if $ad= bc$.

Q2. $\dfrac ab + \dfrac cd = \dfrac{ad+bc}{cd}$.

Q3. $\dfrac ab \cdot \dfrac cd = \dfrac{ac}{bd}$.

How do we justify property Q1? Assume for the moment that rational numbers behave the way we would like them to. If $n$ if a non zero integer, then we want $\dfrac nn$ to act like the number $1$. If that is the case, then we must have $\dfrac ab = \dfrac ab \dfrac nn = \dfrac{an}{bn}$. If we let $c=an$ and $d=bn$ then we find that $ad = bc$. Also we want $\dfrac 0n$ to act like $0$. In which case

\begin{align} \dfrac ab = \dfrac cd &\implies \dfrac{ad}{cd} - \dfrac{bc}{bd} = \dfrac{0}{cd} \\ &\implies \dfrac{ad-bc}{cd} = \dfrac{0}{cd} \\ &\implies ad-bc = 0 \\ &\implies ad = bc \end{align}

Finally, we want to embed the rational numbers $\mathbb Q$ into the set of real numbers $\mathbb R$. The simplest way is to just accept what we want to be true.

R1. For $x,y \in \mathbb R$ with $y \ne 0$ we define $\dfrac xy = x \cdot y^{-1}$.

Note that a lot of books define $y^{-1} = \dfrac 1y$. It isn't too hard to check that Q1, Q2, and Q3 are still true. So R1 is a logically consistent way to embed the rational number into the real numbers.

There are a lot of ways of looking at the fraction $\dfrac 34$. The three most direct ways are:

1. Break an interval that is three units long into four equal pieces and take one of them:

2. Break a unit interval into four equal pieces and take three of them.

3. Divide four into three.

Related to the fraction $\dfrac 34$ is the ratio $3:4$. Two segments are said to have a ratio of $3:4$ if there is a common measure that can be "laid out" exactly 3 times onto the first segment and exactly 4 times onto the second segment.

The idea of a common measure is more primitive than the idea of a fraction. It was once thought that any two line segments had a common measre (were commensurable). We now know that isn't true. A side of a square and a diagonal of that square are not commensurable for example.

Any definition of a fraction should agree with the common ways of looking at a fraction. The three most common ways are:

1. Break an interval that is three units long into four equal pieces and take one of them:

2. Break a unit interval into four equal pieces and take three of them.

3. Divide four into three.

Related to the fraction $\dfrac 34$ is the ratio $3:4$. Two segments are said to have a ratio of $3:4$ if there is a common measure that can be "laid out" exactly 3 times onto the first segment and exactly 4 times onto the second segment.

The idea of a common measure is more primitive than the idea of a fraction. It was once thought that any two line segments had a common measure (were commensurable). We now know that isn't true. A side of a square and a diagonal of that square are not commensurable for example.